If one believes that they believe that p, then do they believe that p ($\text{B} \text{B} p \supset \text{B} p$)?

I was recently thinking about this property and its absence from standard systems of doxastic logic. Systems of doxastic logic rightly do not validate the property $\text{B} p \supset p$. Since they omit this axiom (commonly called the T axiom), $\text{B} \text{B} p \supset \text{B} p$ cannot be simply derived. But although the T axiom should not be valid in a doxastic logic, it is fair to say that the axiom $\text{B} \text{B} p \supset \text{B} p$ should be valid; if one believes that they believe that p, then they do believe that p.

This type of agent is apparently termed a stable reasoner by Raymond Smullyan:

Stable reasoner: A stable reasoner is not unstable. That is, for every p, if it believes Bp then it believes p.

A list of doxastic reasoner types can be found here

Transplication as Implication

In his contribution on partial logic to the Handbook of Philosophical Logic, Stephen Blamey introduces a ‘value gap introducing’ connective named ‘transplication’ to the standard 3-valued partial logic, the Strong Kleene logic. Blamey suggests the possibility of reading the transplication connective as a type of conditional. I was interested to see how the transplication connective fares as a conditional by testing it against a list of inferences concerning conditionals.

Here is the investigation: Transplication as Implication

I have not come across much material concerning transplication. Does anyone else have any other references or ideas?